Paper 2, Section II, F

Probability | Part IA, 2018

(a) Let YY and ZZ be independent discrete random variables taking values in sets S1S_{1} and S2S_{2} respectively, and let F:S1×S2RF: S_{1} \times S_{2} \rightarrow \mathbb{R} be a function.

Let E(z)=EF(Y,z)E(z)=\mathbb{E} F(Y, z). Show that

EE(Z)=EF(Y,Z).\mathbb{E} E(Z)=\mathbb{E} F(Y, Z) .

Let V(z)=E(F(Y,z)2)(EF(Y,z))2V(z)=\mathbb{E}\left(F(Y, z)^{2}\right)-(\mathbb{E} F(Y, z))^{2}. Show that

VarF(Y,Z)=EV(Z)+VarE(Z)\operatorname{Var} F(Y, Z)=\mathbb{E} V(Z)+\operatorname{Var} E(Z)

(b) Let X1,,XnX_{1}, \ldots, X_{n} be independent Bernoulli (p)(p) random variables. For any function F:{0,1}RF:\{0,1\} \rightarrow \mathbb{R}, show that

VarF(X1)=p(1p)(F(1)F(0))2\operatorname{Var} F\left(X_{1}\right)=p(1-p)(F(1)-F(0))^{2}

Let {0,1}n\{0,1\}^{n} denote the set of all 010-1 sequences of length nn. By induction, or otherwise, show that for any function F:{0,1}nRF:\{0,1\}^{n} \rightarrow \mathbb{R},

VarF(X)p(1p)i=1nE((F(X)F(Xi))2)\operatorname{Var} F(X) \leqslant p(1-p) \sum_{i=1}^{n} \mathbb{E}\left(\left(F(X)-F\left(X^{i}\right)\right)^{2}\right)

where X=(X1,,Xn)X=\left(X_{1}, \ldots, X_{n}\right) and Xi=(X1,,Xi1,1Xi,Xi+1,,Xn)X^{i}=\left(X_{1}, \ldots, X_{i-1}, 1-X_{i}, X_{i+1}, \ldots, X_{n}\right).

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